Definition:Jacobi's Iterative Method

Definition

Jacobi's iterative method is an iterative method for solving a system $S$ of simultaneous linear equations.

Let $\mathbf A \mathbf x = \mathbf b$ be a system of simultaneous linear equations expressed in matrix form.

Let $\mathbf A$ be reduced to the form:

$\mathbf A = \mathbf D + \mathbf B$

where $\mathbf D$ denotes the diagonal matrix:

$D_{i i} : = A_{i i}$

where:

$A_{i i}$ and $D_{i i}$ are the elements at the $i$th row and $i$th column of $\mathbf A$ and $\mathbf D$ respectively.

$A_{i i} \ne 0$ for all $i$

One would arrange for $S$ to be expressed in a form as to make the latter statement so.

Let $\mathbf x_0$ be a first approximation to the vector $\mathbf x$.

Jacobi's iterative method generates a sequence of vectors $\mathbf x_1, \mathbf x_2, \ldots$ from the formula:

$\mathbf x_{n + 1} = \mathbf D^{-1} \paren {\mathbf b - \mathbf B \mathbf x_n}$

Suppose $\sequence {x_n}$ converges to the limit $\mathbf x$.

Then $\mathbf x$ is the solution to $\mathbf A \mathbf x = \mathbf b$.

Motivation

Jacobi's iterative method was designed for scenarios in which the diagonal elements of the matrices involved are relatively large.

Hence matrices which are diagonally dominant are particularly susceptible to this treatment.

Also known as

Jacobi's iterative method is also known just as the Jacobi method.

Also see

• Results about Jacobi's iterative method can be found here.

Source of Name

This entry was named for Carl Gustav Jacob Jacobi.

Historical Note

Carl Gustav Jacob Jacobi published his iterative method in $1845$.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Jacobi method